A333161 Triangle read by rows: T(n,k) is the number of k-regular graphs on n unlabeled nodes with half-edges.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 8, 12, 8, 4, 1, 1, 4, 10, 24, 24, 10, 4, 1, 1, 5, 17, 70, 118, 70, 17, 5, 1, 1, 5, 24, 172, 634, 634, 172, 24, 5, 1, 1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1, 1, 6, 50, 1530, 35500, 187990, 187990, 35500, 1530, 50, 6, 1

Offset

0,5

Comments

A half-edge is like a loop except it only adds 1 to the degree of its vertex.

T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333159.

T(n,k) is the number of simple graphs on n unlabeled vertices with every vertex degree being either k or k-1.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..230

Formula

T(n,k) = T(n, n-k).

Example

Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  2,   1;
  1, 3,  3,   3,    1;
  1, 3,  4,   4,    3,    1;
  1, 4,  8,  12,    8,    4,    1;
  1, 4, 10,  24,   24,   10,    4,   1;
  1, 5, 17,  70,  118,   70,   17,   5,  1;
  1, 5, 24, 172,  634,  634,  172,  24,  5, 1;
  1, 6, 36, 525, 4428, 9638, 4428, 525, 36, 6, 1;
  ...
The a(2,1) = 2 adjacency matrices are:
  [0 1]  [1 0]
  [1 0]  [0 1]
.
The A(4,2) = 3 adjacency matrices are:
  [0 0 1 1]   [1 1 0 0]   [1 1 0 0]
  [0 0 1 1]   [1 1 0 0]   [1 0 1 0]
  [1 1 0 0]   [0 0 1 1]   [0 1 0 1]
  [1 1 0 0]   [0 0 1 1]   [0 0 1 1]

Crossrefs

Columns k=0..3 are A000012, A004526(n+2), A186417, A333163.

Row sums are A333162.

Central coefficients are A333166.

Cf. A051031, A333157, A333159.

Sequence in context: A103691 A103441 A081206 * A156044 A180980 A275298

Adjacent sequences: A333158 A333159 A333160 * A333162 A333163 A333164

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 11 2020

Status

approved